7 replies to this topic

[Tom Polakis]

A couple years ago, I took a series of images of a total lunar eclipse, and then set it into motion with a time-lapse sequence.  What struck me was the 'wobble' of the moon during the four-hour period between the beginning and end of the eclipse.  The animated GIF below shows this effect, which is due to parallax.  During those four hours, the earth's rotation moved the observer moved a transverse distance of 4000 miles, which amounts to nearly 1 degree of parallax.

 

I have always seen libration defined as being due to the moon's elliptical orbit.  So here's my semantic question: is the effect of parallax due to earth's rotation considered a part of libration?  It does help us "see around the corner" of the moon, but I don't think it is.

 

Tom

 

[Bonco]

I'll take a stab at it and won't complain if someone shoots me down. Parallax is what's seen in your photo's. In four hours libration would be minuscule.

Longitudinal libration is caused by the changing speed  of the moon in it's orbit. This is because it's orbit is elliptical. Closer to earth = faster. Latitudinal libration is caused by the fact it's equator  doesn't exactly coincide with it's orbital plane. Bill

Edited by Bonco, 15 March 2016 - 04:48 PM.

[Centaur]

Thanks, Tom, for your fine photographic demonstration of "diurnal libration" due to the effects of parallax. While it is not given consideration for a chart of geocentric based lunar librations, it is indeed one of four types of libration that must be taken into account by a topocentric observer. It is one of the three types of libration that are just geometric effects and referred to as "optical librations". There is also an actual "physical libration", but its effect for an observer is miniscule compared to the others.

[Centaur]

I could add that the other two types of "optical librations" are "longitudinal" and "latitudinal" as Bonco described. I didn't realize he had replied while I was composing my earlier post.

 

To be more precise, the orbital consideration that produces longitudinal libration is due to the Moon's "eccentric" orbit and not much at all due to it being "elliptical". Yes, eccentricity has a complex mathematical correlation with ellipticity, but the Moon's orbit has such low ellipticity that it is essentially circular. The far more noticeable eccentricity is a measure of the degree to which the Earth is off from the center of the Moon's orbit.

 

That eccentricity results in variances in the Moon's orbital speed, especially its angular speed relative to Earth, while its rotational speed around its axis is almost constant. Hence the apparent longitudinal libration. 

 

A perfectly circular orbit has an eccentricity of 0 and an ellipticity of 1. The eccentricity of the Moon's orbit is the ratio of the distance between the Earth's center and the orbital center relative to the semi-major axis. Ellipticity is the ratio of the semi-major axis relative to the semi-minor axis.

 

Below is a graphic I designed to illustrate the dominance of eccentricity over ellipticity.

 

 

Edited by Centaur, 16 March 2016 - 01:00 PM.

[Bonco]

Well at least Curt didn't totally blow me out of the water. Thanks Curt for the better response. 

Bill

[Tom Polakis]

Thanks, Curt and Bill.  Great explanations.

 

Tom

[Codbear]

Hi Tom,

 

That not only was a great question you posed, but a fantastic GIF of the moon going through diurnal libration as Bill and  Curt described so succinctly.  The only bad part about that GIF is it's giving me some really painful flashbacks of some awful disco parties I went to in the '70s! lol.

 

If I may add one more comment about Diurnal Libration. Unlike Longitudinal and Latitudinal Libration, which as stated is primarily a function of a combination of the eccentricity and elliptical nature of the moon's orbit, Diurnal Libration is greatly affected by WHERE you are on earth.

 

Your calculation of 4,000 miles for the transverse distance an observer travelled during the 4 hours of the lunar eclipse, resulting in an approximate 1 degree of parallax, would only be experienced at the equator. As you travel farther away from the equator, the transverse distance (or baseline) used for the parallax calculation becomes shorter and shorter, until it's the size of a gnat's brain at the poles ( and mine sometimes according to my wife!!!). To calculate the baseline for any given latitude, multiply the 4,000 mile radius of the Earth by the sine of the complement of your latitude. 

 

As an example, where you are in Tempe is about 33 degrees latitude, so sin57*4,000 = 3,355 miles.  Since the length of the baseline is directly proportional to the observed parallax, the approximate 1 degree of observed lunar parallax at the equator would only be about .8 degrees. Where I am, about half an hour north of San Francisco, its about 3/4 of a degree, and in Anchorage it's a tiny bit less than half a degree.

 

As a final thought, and I'd enjoy hearing feedback from any or all of you. I thought I had read somewhere years ago that a small portion of libration is also due to the uneven gravitational attraction the earth has on the moon due to the earth's varying density, thus slightly changing the orbital speed of the moon. While this definitely affects the orbital speed (albeit it slightly) of something like the ISS, I don't think the libration of something as far away as the moon would be observable.

[Tom Polakis]

Your calculation of 4,000 miles for the transverse distance an observer travelled during the 4 hours of the lunar eclipse, resulting in an approximate 1 degree of parallax, would only be experienced at the equator. As you travel farther away from the equator, the transverse distance (or baseline) used for the parallax calculation becomes shorter and shorter...

 

Thanks.  I should have thought of that.

 

I should add that the middle of the two frames was very close to local midnight.  If the sequence spanned from, say, sunset to four hours after sunset, the parallax would have been much less.

 

Tom